Answer
The set of function is linearly independent on $(-\infty, \infty)$ as $W$ is different from zero.
Work Step by Step
The Wronskian of the function can be defined as:
$W[e^{ax} \cos b x, e^{ax} \sin b x, xe^{ax} \cos bx , xe^{ax} \sin bx]=4b^4 e^{4ax}$
Since, $W$ can be different from zero, so the set of function is linearly independent on $(-\infty, \infty)$.
